## Product Description

The fact that you chose to read this book makes it likely that you might have heard of Kurt Gödel,2 the greatest logician since Aristotle,3 whose arguably revolutionary discoveries influenced our views on mathematics, physics, and philosophy, comparable only to the discoveries of quantum mechanics.Well, even if you have not heard of him I want to start by rephrasing his famous theorem:

Mathematics is inexhaustible!

Notwithstanding the lack of a definition of what mathematics is, that still sounds wonderful, doesn’t it? At this point, you may not fully understand the meaning of this “theorem” or appreciate its significance for mathematics and philosophy. You may even disagree with it, but I suppose you would agree with me that mathematics is the study of abstract structures with enormous applications to the “real world.” Also, wouldn’t you agree that the most impressive features of mathematics are its certainty, abstractness, and precision? That has always been the case, and mathematics continues to be a vibrant, constantly growing, and definitely different discipline from what it used to be. I hope you would also agree (at least after reading this book) that it possesses a unique beauty and elegance recognized from ancient times, and yet revealing its beauty more and more with/to every new generation of mathematicians. Where does it come from? Even if you accept the premise that it is a construct of our mind, you need to wonder how come it represents/reflects reality so faithfully, and in such a precise and elegant way. How come its formalism matches our intuition so neatly? Is that why we “trust” mathematics (not mathematicians) more than any other science; indeed, very often we define truth as a “mathematical truth” without asking for experimental verification of its claims? So, it is definitely reasonable to ask at the very beginning of our journey (and we will ask this question frequently as we go along): Does the world of mathematics exist outside of, and independently of, the physical world and the actions of the human mind? Gödel thought so. In any case, keep this question in mind as you go along – it has been in the minds of mathematicians and philosophers for centuries.

The set theory that we start with comes as a culmination of 2000 years of mathematics, with the work of the German mathematician George Cantor4 in the 1890s. As much as the inception of set theorymight have had (apparently) modest beginnings, there is virtually no mathematical field in which set theory doesn’t enter as the very foundation of it. And it does it so flawlessly, so naturally, and in such a “how-could-it-be-otherwise” way, that one wonders why it took us so long to discover it. And arguably, there is no concept more fundamental than the concept of the set. (Indeed, try to answer the question: What is a real number without referring to set theory?) Be it as it may, now we have it. We start our journey through the “Principles,” with the basic formalism of set theory.